G. Tardos | Searcharxiv

arXiv2013

Optimal information rate of secret sharing schemes on trees

The information rate for an access structure is the reciprocal of the load of the optimal secret sharing scheme for this structure. We determine this value for all trees: it is 1/(2-1/c), where c is the size of the largest core of the tree. A subset of the vertices of a tree is a core if it induces a connected subgraph and for each vertex in the subset one finds a neighbor outside the subset. Our result follows from a lower and an upper bound on the information rate that applies for any graph and happen to coincide for trees because of a correspondence between the size of the largest core and a quantity related to a fractional cover of the tree with stars.

cs.CR↗

arXiv2007

Graph coloring with no large monochromatic components

For a graph G and an integer t we let mcc_t(G) be the smallest m such that there exists a coloring of the vertices of G by t colors with no monochromatic connected subgraph having more than m vertices. Let F be any nontrivial minor-closed family of graphs. We show that \mcc_2(G) = O(n^{2/3}) for any n-vertex graph G \in F. This bound is asymptotically optimal and it is attained for planar graphs. More generally, for every such F and every fixed t we show that mcc_t(G)=O(n^{2/(t+1)}). On the other hand we have examples of graphs G with no K_{t+3} minor and with mcc_t(G)=Ω(n^{2/(2t-1)}). It is also interesting to consider graphs of bounded degrees. Haxell, Szabo, and Tardos proved \mcc_2(G) \leq 20000 for every graph G of maximum degree 5. We show that there are n-vertex 7-regular graphs G with \mcc_2(G)=Ω(n), and more sharply, for every ε>0 there exists c_ε>0 and n-vertex graphs of maximum degree 7, average degree at most 6+εfor all subgraphs, and with mcc_2(G)\ge c_\eps n. For 6-regular graphs it is known only that the maximum order of magnitude of \mcc_2 is between \sqrt n and n. We also offer a Ramsey-theoretic perspective of the quantity \mcc_t(G).

math.CO↗